Using UV Mapping on an Equirectangular Projection
Two bake solutions.

Bake from the original object

A solution can be baking the model from this answer on to a copy of the mesh.

To do that:


From the original mesh, enter edit mode and unwrap it (the way you want, smart UV project will do).
In object mode duplicate the object, so that both original and copy will have a UV map which will be needed to bake even if the UV map of the original mesh is not directly used.


The UV map I have created is the following:




Create a new material for the copied mesh.
Add a unconnected image texture in it, call it for instance 'bake' and save it to disk.


Like so:



Now the bake setting:


Use Cycles and if possible GPU
Set the bake mode to diffuse (we want the color)
Influence color only (for the same reason)
Selected (original mesh) to active (target mesh)
Ray distance to some little amount to avoid overlapping effects between the two meshes




Now make sure the two objects are at the exact same position.

Select the original mesh, shift select the target, and use the bake button above.

Here are the resulting meshes:



(I cannot share the blend file as too large for this site).

Bake from the environment texture

Another possibility is to bake from the environment texture.

As we know that the original texture is an equirectangular projection, we can set it as environment texture.

Here too the object is previously unwrapped the way you want.

The bake setting is simple (the object will have to be at the world center):



Now we have to setup the environment texture. As for your previous question the tricky part is to find the good mapping in this context (180 degrees and inverted axis):



Once done, select your object and hit the "bake" button.

The result is similar: